Finite Group DatabasebetaFinite Group Databasebeta| order | 160 |
| isoclinism | isoclinic class of AGL(1,𝔽5) |
| automorphism | Aut(G) = C8×AGL(1,𝔽5) (order 160) · Inn(G) = AGL(1,𝔽5) · Out(G) = C8 |
| holomorph | G ⋊ Aut(G) = #72242339 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C5 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | AGL(1,𝔽5) |
| commutator, hypocenter | C5 | C32 |
| Fitting | C5×C8 | C4 |
| socle, prime radical | C2×C5 | C16 |
| cosocle | #672 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C8 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(2,3,4,5)(6,7,8,12,9,13,16,22,10,14,17,23,19,25,28,32,11,15,18,24,20,26,29,33,21,27,30,34,31,35,36,37);(1,2,3,5,4)Polycyclic presentation:
2^5,5|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^2,f^b=f^440 conjugacy classes — view the full character table.
| maximal subgroup of | D5⋊C32, #64842, C2×#3928, … |
| maximal normal subgroup of | D5⋊C32, #64842, C2×#3928, … |
| maximal quotient of | #64665, C2×#3928, C3×#3928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(160, 3);